Chapter 14: Q29P (page 687)

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.
$\frac{e^{2 z}}{4 c o s h z - 5}$ at $z = l n 2$

Short Answer

Expert verified

The residue of the function at $z = \ln 2$ is $R (\ln 2) = \frac{4}{3}$

Step by step solution

Determine the formula

Simple pole at the residue is given by the function as:

$R [f (z)] = \lim_{z \to z_{0}} (z - z_{0}) f (z)$

Determine the residue of the simple pole

Consider the given function as:

$f (z) = \frac{e^{2 z}}{4 \cos h z - 5}$

Rewrite the function as:

$\begin{array}{rcl} f (z) & = & \frac{q (z)}{p (z)} \\ = & \frac{e^{2 z}}{4 \cos h z - 5} \end{array}$ …… (1)

At $\begin{array}{rcl} z & = & \ln 2 \end{array}$

The function has simple pole at $\begin{array}{rcl} z & = & \ln 2 \end{array}$ and the residue of simple pole is given by:

$\begin{array}{rcl} R (z_{0}) & = & \lim_{z \to z_{0}} (z - z_{0}) f (z) \\ = & \frac{q (z_{0})}{p (z_{0})} \end{array}$ ……. (2)

Determine the residue of the function as:

Substitute the values in equation (2) as follows:

$\begin{array}{rcl} R (\ln 2) & = & \frac{e^{2 \ln 2}}{4 \sin h (\ln 2)} \\ = & \frac{1}{\sin h (\ln 2)} \\ = & \frac{4}{3} \end{array}$

Therefore, the residue of a function at $\begin{array}{rcl} z & = & \ln 2 \end{array}$ is $\begin{array}{rcl} R (\ln 2) & = & \frac{4}{3} \end{array}$ .

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Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.
$\frac{e^{2 z}}{4 c o s h z - 5}$ at $z = l n 2$

Short Answer

Step by step solution

Determine the formula

Determine the residue of the simple pole

Determine the residue of the function as:

One App. One Place for Learning.

Most popular questions from this chapter

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Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.e2z4coshz-5atz=ln2

Short Answer

Step by step solution

Determine the formula

Determine the residue of the simple pole

Determine the residue of the function as:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fields in Physics

Mechanics and Materials

Fluids

Measurements

Geometrical and Physical Optics

Electricity and Magnetism

Study anywhere. Anytime. Across all devices.

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.
$\frac{e^{2 z}}{4 c o s h z - 5}$ at $z = l n 2$